On the Expressions of Coplanarity and Homoconicism. 379 



companied the oxalic acid, I could not determine from want of 

 material. The presence of oxalic acid however is distinctly indi- 

 cated by the above reactions. They likewise show the presence 

 of chloride of sodium, potash, sulphuric acid and magnesia. 



In comparing this secretion of the leaves of the Ice-plant with 

 the fluid in the ascidia of Nepenthes, we find a material difference 

 in their respective compositions, as will be seen by the annexed 

 table, which exhibits the composition of both fluids : — 



Coinposition of the watery secretion 

 of the leaves of Mesembryanthe- 

 mum crystallinum. 



Organic matter (albumen, oxalic 

 acid, &c.). 



Chloride of sodium. 



Potash. 



Magnesia. 



Sulphuric acid. 



Composition of the fluid in the 

 ascidia of Nepenthes. 



Organic matter, chiefly mahc and a 



Uttle citric acid. 

 Chloride of potassium. 

 Soda. 

 Lime. 

 Magnesia. 



XLVI. On the Qtiaternion Expressions of Coplanarity and 

 Homoconicism. By William Spottiswoode, M.A.^ of 

 Balliol College^ Oxford*. 



THE following investigations relate to certain theorems 

 given by Sir W. R. Hamilton in vol. xxix. of this 

 Journal. Adopting the notation of the original papers, the 

 equation 



S.uu^Uc^ — O (1.) 



(where suffixes are used instead of accents) is equivalent to 



%» 2J] , Set 



= 0, 



(2.) 



which may be replaced by 



rt«-f-aiai + «2a2=0; (3.) 



because, there being no linear relation between e, j, ^, this 

 last is equivalent to the system 



ax -^ a-^x^-\- ac^ci=^^ 



«y + «iJ/i + «2i/2=0 r • .... (4.) 



«2; + «! iZj + ff 2 2^2 = -J 



and (2.) or (1.) is the result of these. Hence (1.) and (3.) are 

 alike conditions of coplanarity. 

 Again, if 



/3i = V.V.«i«2-V.«4«5k .... (5.) 

 ^2 = V.V.«2«3.V.«5aJ 



* Communicated by the Author. 

 2 C 2 



